The present invention relates generally to function graphing and, more specifically, to automatic determination of optimum graph characteristics.
Mathematical functions can be classified as either algebraic or transcendental (nonalgebraic). Algebraic functions are expressions that involve only the algebraic operations of addition, subtraction, multiplication, division, raising to an integer power, and extracting an odd number root or even number root. Elementary transcendental functions include exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic trigonometric, inverse hyperbolic trigonometric, tower functions, as well as combinations of these functions together with algebraic (including polynomial and rational) functions into piecewise-defined functions. In terms of graphing, a key difference that distinguishes both algebraic functions and transcendental functions that do not include any of the six trigonometric functions from other transcendental functions is that the former functions have a finite number of points of mathematical interest such as: intercepts, extrema, points of inflection, holes, vertical asymptotes, horizontal asymptotes, slant asymptotes, points of discontinuity, isolated points, vertical tangents, vertical cusps, and corners. This means that graphs of algebraic functions and transcendental functions that do not include any of the six trigonometric functions are capable of being displayed in a complete graph—one that includes all their points of mathematical interest. Other transcendental functions, on the other hand, including the six trigonometric functions, can have an infinite number of points of mathematical interest.
Software packages and many graphing calculators use algorithms that could display complete graphs of functions if the selection of function-specific graphing settings such as x-dimension values and y-dimension values were appropriate. Further, if all the important mathematical features, together with standard auxiliary graphics for illuminating singularities (holes, discontinuities, isolated points, and the various kinds of asymptotes), could be provided for in the function-specific settings, significant features could be more clearly apparent.
What is needed is a system and method to provide an electronic graph of a function in which important mathematical features of the function, including singularities, are clearly apparent, for example, but not limited to, within an optimum viewing window, for example, but not limited to, on a graphing calculator, a personal computer display, a handheld computer, or other type of electronic display. Further, a system and method are needed to provide an efficient way for a user to view mathematical features of a user-selected function. Still further, the use of traditional auxiliary graphics for illuminating singularities (dashed lines to designate asymptotes, small empty circles to designate excluded curve endpoints, and small filled circles to designate included curve end points or isolated points) can help users to better identify unusual features of a given function from its graph, and, moreover, mathematical analyses of a given function reported with numerical results in both exact (symbolic) forms, which users typically get when obtaining solutions by hand, as well as approximate forms can help users to review calculation results.